AS/A2 Pure 4 Pure 5

P6 Topic 1: Complex numbers
Circle loci 2: |z - z1| = a
The locus of z when |z - z1| = a, where a is a fixed constant and z1 is a fixed known complex number. If z=x+iy is represented on an Argand diagram by the vector going from the origin to the point P with coordinates (x,y), and z1 = x1+iy1 is a known complex number represented by the vector OQ where Q has coordinates (x1,y1), then the equation |z - z1| = a requires the distance of P from Q to be a units. So the locus is a circle, centre (x1,y1), with radius a units. In the diagram below you can vary the value of a. and the vector z1. Clearly, P must lie on the circle shown.

Summary |z| is the modulus of the complex number z. If z is represented on an Argand diagram by the vector OP then |z| is the length of the vector OP. If z1 is another complex number represented by OQ then |z - z1| is the distance from P to Q. Cabri Geometre interactive geometry.